18-05-2012, 12:37 PM
Ziegler–Nichols method
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Ziegler-Nichols Method:
First, note whether the required proportional control gain is positive or negative. To do so, step the input u up (increased) a little, under manual control, to see if the resulting steady state value of the process output has also moved up (increased). If so, then the steady-state process gain is positive and the required Proportional control gain, Kc, has to be positive as well.
1. Turn the controller to P-only mode, i.e. turn both the Integral and Derivative modes off.
2. Turn the controller gain, Kc, up slowly (more positive if Kc was decided to be so in step 1, otherwise more negative if Kc was found to be negative in step 1) and observe the output response. Note that this requires changing Kc in step increments and waiting for a steady state in the output, before another change in Kc is implemented.
3. When a value of Kc results in a sustained periodic oscillation in the output (or close to it), mark this critical value of Kc as Ku, the ultimate gain. Also, measure the period of oscillation, Pu, referred to as the ultimate period. ( Hint: for the system A in the PID simulator, Ku should be around 0.7 and 0.8 )
4. Using the values of the ultimate gain, Ku, and the ultimate period, Pu, Ziegler and Nichols prescribes the following values for Kc, tI and tD, depending on which type of controller is desired:
What is the Ziegler-Nichols Rule?
The Ziegler-Nichols rule is a heuristic PID tuning rule that attempts to produce good values for the three PID gain parameters:
1. Kp - the controller path gain
2. Ti - the controller's integrator time constant
3. Td - the controller's derivative time constant
given two measured feedback loop parameters derived from measurements:
1. the period Tu of the oscillation frequency at the stability limit
2. the gain margin Ku for loop stability
with the goal of achieving good regulation (disturbance rejection).
When Does the Ziegler-Nichols Rule Work?
Tuning rules work quite well when you have an analog controller, a system that is linear, monotonic, and sluggish, and a response that is dominated by a single-pole exponential "lag" or something that acts a lot like one.
Actual plants are unlikely to have a perfect first-order lag characteristic, but this approximation is reasonable to describe the frequency response rolloff in a majority of cases. Higher-order poles will introduce an extra phase shift, however. Even if they don't affect the shape of the gain rolloff much, the phase shift matters a lot to loop stability. You can't depend upon a single "lag" pole to match both the amplitude rolloff and the phase shift accurately.
So the Ziegler-Nichols model presumes an additional fictional phase adjustment that does not distort the assumed magnitude rolloff. At the stability margin, there is a 180 degree phase shift around the feedback loop (Nyquist's stability criterion). A first order lag can contribute no more than 90 degrees of that phase shift. The rest of the observed phase shift must be covered by the artificial phase adjustment. The phase adjustment is presumed to be a straight line between zero and the critical frequency where 180 degrees of phase shift occurs. A "straight line" phase shift corresponds to a pure time delay. Is this consistent with the actual phase shifts? Well, probably not, so hope for the best.